202 iX ANALYTIC FUNCTIONS

gk(u) = £ 6<*V, Gjfe(w) = £ I*?' !«*• On the other hand' let (av^v) be a power

# /i

series In p variables with coefficients in E, which is absolutely summable
in a polydisk P of Kp, of center 0 and radii rk(l^k^p). If, in a monomial
zv = znl1 ...zn/, we replace "formally" each zk by the power series gk(u\
we are led to take the formal "product" of n^ + n2 + • • • + np series, i.e.
to pick a term in each of the n± + • - • + np factors, to take their product
and then to "sum" all terms thus obtained. We are thus led to consider,
for each v = (X, n2 , . . . , np) the set Av of all finite families Qj,kJ) = p where
fikj e N*, k ranges from 1 to p, and for each k, j ranges from 1 to nk\ to such
a p we associate the element

nk

With these notations:

(9.2.1) Suppose sl9...,sq are q numbers >0 satisfying the conditions
Gjtfo,..., sq) < rk for l^k^p. Then, for each u in the open polydisk
S c Kq of center 0 and radii st (1 < i" < q), the family (tp(u)) (where p ranges
through the denumerable set of indices A = (j Av) is absolutely summable,

and iff(z) = £ avzv, its sum is equal toftg^u), g2(u), ..., gp(u)).

In other words, under the conditions Gk(sl9 ..., sq) < rk (l^
"substitution" of the series gk(u) for zk (1 < k </?) in the series / yields
an absolutely summable family, even before all the terms tp(u) having the
same degrees in ul9..., uq have been gathered together.

To prove (9.2.1), we need only prove that the family (tp(u)) is absolutely
summable; that its sum is/(^(w)?..., gp(u)) follows by application of the
associativity theorem (5.3.6) to the subsets Av of A, and by using (5.5.3),
which shows that £ tp(u) is equal to a^g^u))"1 ... (gp(u))np. To prove the

peAv

family (tp(u)) (peA) is absolutely summable, we apply (5.3.4). For any
finite subset B of A, we have, by (5.3.5) and (5.5.3)

Z uncoil < KB • (Gi(s19..., sjr... (QP(sl9..., s,))"'

p 6 B n Av

and by assumption, the right-hand side of that inequality is the element of
index v of an absolutely summable family; hence the result.

Write tp(u) = cpu\ with A = (^,..., ^), ^ = f V mkji (if we have

*=i j~i
Hkj = (w*ji» • • •»^> ))• From (9.2.1) and (5.3.5) it follows (taking all the ut Give an example in which R" > RRX.